Question

Find the following limits or state that they do not exist. Assume \(a, b, c,\) and k are fixed real numbers. $$\lim _{x \rightarrow 0} \frac{\sin 2 x}{\sin x}$$

Short answer

Question: Determine the limit of the function \(f(x) = \frac{\sin 2x}{\sin x}\) as \(x\) approaches 0. Answer: The limit of the given function, as x approaches 0, is 2.

Step-by-step solution

Identify the given function

We are given the function: $$f(x) = \frac{\sin 2x}{\sin x}$$

Use the trigonometric identity for sine of double angle

We will use the double angle identity for sine which is: $$\sin 2x = 2 \sin x \cos x$$ Replace the \(\sin 2x\) in our function with this identity: $$f(x) = \frac{2 \sin x \cos x}{\sin x}$$

Simplify

Notice that we can cancel out \(\sin x\) in both the numerator and denominator: $$f(x) = 2 \cos x$$

Find the limit as x approaches 0

Now our function is simplified to \(f(x) = 2 \cos x\). We can easily find the limit as x approaches 0 using the direct substitution method: $$\lim_{x \rightarrow 0} 2 \cos x = 2 \cos (0) = 2(1) = 2$$ The limit of the given function, as x approaches 0, is 2.

Key concepts

Trigonometric Identities

Trigonometric identities are very useful when solving calculus problems involving limits. These identities, such as the double angle formula, help us express complex trigonometric expressions in a simpler form. The double angle identity for sine is particularly handy here. It states that \( \sin 2x = 2 \sin x \cos x \). By knowing this formula, we can rework the expression \( \frac{\sin 2x}{\sin x} \). This identity turns our original expression into something that's much easier to manage. It allows us to break down problems involving angles into simpler components by using basic trigonometric functions like sine and cosine. Applying these identities reduces complication, making it easier for the next steps in our limit calculation.

Limit Simplification

The limit simplification process involves reducing a complex expression into a form where the limit can be easily determined. In our exercise, after applying the trigonometric identity \( \sin 2x = 2 \sin x \cos x \), we get the expression \( \frac{2 \sin x \cos x}{\sin x} \). At this point, simplification is key. By simplifying, we can cancel out the \( \sin x \) terms in the numerator and the denominator.

This simplification transforms our function into \( 2 \cos x \). This process of canceling terms is crucial because it removes any undefined conditions, like dividing by zero, as we later make direct substitution to find the limit. Simplifying expressions often makes it clear what happens as the variable approaches a certain value, in our case, zero. This step sets the stage for using the direct substitution method effectively.

Direct Substitution Method

The direct substitution method is a straightforward yet powerful technique for evaluating limits. Once we have successfully simplified the expression, we can directly substitute \( x = 0 \) into the function. For our example, the simplified function is \( 2 \cos x \).

Using the direct substitution method, we replace \( x \) with 0, yielding \( 2 \cos (0) \). Since \( \cos(0) = 1 \), it simplifies to \( 2 \times 1 = 2 \).

• Ensure the expression is simplified to avoid undefined results, like division by zero.
• Verify that substitution results do not lead to indeterminate forms such as \( \frac{0}{0} \).

In cases where a function is not continuous at the point of interest, or we face indeterminate forms, other methods might be necessary. However, for continuous functions like our example, this method provides a quick and accurate solution.